AirPlace Kyriakos Georgiou Athina Paphitou Maria Christodoulou 1EPL371 – Systems Programming.
1 On the price of anarchy and stability of correlated equilibria of linear congestion games By...
-
date post
20-Dec-2015 -
Category
Documents
-
view
215 -
download
2
Transcript of 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By...
![Page 1: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/1.jpg)
1
On the price of anarchy and stability of correlated equilibria of linear congestion games
By George Christodoulou Elias Koutsoupias
Presented by Efrat Naim
Part of slides taken from George Christodoulou and Elias Koutsoupias web site
![Page 2: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/2.jpg)
2
Agenda
Congestion Games
An example
Definitions
Bounds for correlated Price of stability of congestions games
Bounds for correlated price of anarchy of congestion games
Related Work
Results
![Page 3: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/3.jpg)
3
Congestion Games
Introduced in [Rosenthal, 1973]
Each player has a source and destination.
Pure strategies are the path from source to destination
The cost on each edge depends on the number of the players using it.
d c
ba
![Page 4: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/4.jpg)
4
Congestion Games
N players
M facilities (edges)
Pure strategy (path) is a subset of facilities.Each player can select among a collection of pure strategies (pure strategy set)
Cost of facility depends on the number of players using it
The objective of each player is to minimize its own total cost
Pure strategy profile s = (s1,……..sN)
![Page 5: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/5.jpg)
5
An Example
ed
c
ab
![Page 6: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/6.jpg)
6
An Example
From a to c
ed
c
ab
![Page 7: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/7.jpg)
7
An Example
From a to c
ed
c
ab
![Page 8: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/8.jpg)
8
An Example
From a to c
ed
c
ab
![Page 9: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/9.jpg)
9
An Example
From e to c
ed
c
ab
![Page 10: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/10.jpg)
10
An Example
From e to c
ed
c
ab
![Page 11: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/11.jpg)
11
An Example
Nash Equilibrium
Player 1 has cost 1+1=2Player 2 has cost 1+1 =2
ed
c
ab
![Page 12: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/12.jpg)
12
An Example
Another Nash Equilibrium
Player 1 has cost 2+1+1=4Player 2 has cost 2+1+1=4
ed
c
ab
![Page 13: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/13.jpg)
13
Mixed Strategy
A mixed strategy for a player is a probability distribution over its pure strategy set.
Mixed strategy profile p = (p1,…….pN)
ed
c
ab
1/2
1/2
![Page 14: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/14.jpg)
14
Correlated Strategy
A correlated strategy q for a set of players is any probability distribution over the set S = X i€N Si
ed
c
ab
ed
c
ab
1/2 1/2
1/2 0
0 1/2
L
R
L R
1
2
![Page 15: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/15.jpg)
15
Correlated Equilibrium
Introduced in [Auman,1974]
Consider a mediator that makes a random experiment with a probability distribution q over the strategy space S.
q is common knowledge to the players
The mediator, with respect to the outcome s€S , announces privately the strategy si to the player i.
Player i is free to obey or disobey to the mediator’s recommendation, with respect to his own profit.
Player i doesn’t know the outcome of the experiment
If the best for every player is to follow mediator’s recommendation , then q is a correlated equilibrium.
![Page 16: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/16.jpg)
16
Correlated Equilibrium
Cost for player i for pure strategy A is
ne(A) = number of players using e in A
Given a correlated strategy q, the expected cost of a player i€N is
A correlated strategy q is a correlated equilibrium if it satisfies the following
![Page 17: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/17.jpg)
17
Price of Anarchy
Price of anarchy
A social cost (objective) of a pure strategy profile A is the sum of players costs in A:
and
![Page 18: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/18.jpg)
18
Price of Anarchy
ed
c
ab
Player 1 has cost 2+1+1=4
Player 2 has cost 2+1+1=4
PoA = (4+4)/ (2+2) = 2
![Page 19: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/19.jpg)
19
Price of Stability
Price of stability
![Page 20: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/20.jpg)
20
Price of Stability
ed
c
ab
Player 1 has cost 1+1=2
Player 2 has cost 1+1=2
PoS = (2+2)/ (2+2) = 1
![Page 21: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/21.jpg)
21
PoS and PoA of congestion games
By PoS(PoA) for a class of games, we mean the worst case Pos(PoA) over this class.
UpperBound: must hold for every congestion game
LowerBound: Find such a congestion game
![Page 22: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/22.jpg)
22
Correlated PoS – Upper Bound
We consider linear latencies
fe(x) = aex+be
Lemma 1:For every pair of non negative integers it holds
![Page 23: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/23.jpg)
23
Correlated PoS – Upper Bound
Theorem 1: Let A be a pure Nash equilibrium and P be any pure
strategy profile such that P(A)<= P(P), then SUM(A)<=8/5AUM(P)
Where P is the potential of strategy profile
This show that the correlated price of stability is at most 1.6
![Page 24: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/24.jpg)
24
Correlated PoS – Upper Bound
Proof:
Let X be a pure strategy profile X=(X1,………XN)
From the potential inequality
![Page 25: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/25.jpg)
25
Correlated PoS – Upper Bound
A is Nash equilibrium so
Summing for all the players we get
Adding the two inequalities and use Lemma 1
![Page 26: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/26.jpg)
26
Price of Stability - Lower Bound
Dominant Strategies:Each player prefers a particular strategy (dominant), no matter
what the other players will choose.
Dominant Strategies Nash Equilibrium Correlated Equilibrium
![Page 27: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/27.jpg)
27
Price of Stability - Lower Bound
Theorem 2:There are linear congestion games whose dominantequilibrium have price of stability of the SUM social cost approachingas the number of players N tends to infinity.
So this holds for correlated equilibrium.
![Page 28: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/28.jpg)
28
Price of Stability - Lower Bound
A strategies type P strategies
(equilibrium) (optimal social cost)1
2
3
N-1
N
1
2
3
N-1
N
![Page 29: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/29.jpg)
29
Price of Stability - Lower Bound
A strategies type P strategies
(equilibrium) (optimal social cost)1
2
3
N-1
N
1
2
3
N-1
N
![Page 30: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/30.jpg)
30
Price of Stability - Lower Bound
A strategies type P strategies(equilibrium) (optimal social
cost)
1
2
3
N-1
N
1
2
3
N-1
N
![Page 31: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/31.jpg)
31
Price of Stability - Lower Bound
We will fix and m such that in every allocation (A1,……AK,PK+1……PN), players
prefer their Ai strategies.
In order to be dominant (A1, ……. AN)
![Page 32: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/32.jpg)
32
Price of Stability - Lower Bound
And it is satisfied by
For , the price of anarchy
tends to as N tends to
infinity.
![Page 33: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/33.jpg)
33
Correlated PoA- Upper Bound
Theorem 4:The correlated price of anarchy of the average
social cost is 5/2.
Lemma 2:For every non negative integers :
![Page 34: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/34.jpg)
34
Correlated PoA- Upper Bound
Proof: Let q be a correlated equilibrium and P be an
optimal allocation.
Summing for all players
The optimal cost is
![Page 35: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/35.jpg)
35
Correlated price of anarchy
![Page 36: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/36.jpg)
36
Correlated price of anarchy – cont.
Sum over all players i
We finally obtain
![Page 37: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/37.jpg)
37
Asymmetric weighted games
Theorem 6:For Linear weighted congestions games, the correlated price of anarchy of the total latency is at most
Lemma 3: For every non negative real :
Build so satisfy
Achieved by
![Page 38: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/38.jpg)
38
Asymmetric weighted games
Proof:
Q – correlated equilibrium,P - optimal allocation e(s) - total load on the facility e for allocation s
multiply with i
Nash inequality
![Page 39: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/39.jpg)
39
Asymmetric weighted games
And for all players:
PoA = C(q)/ C(P) = (3+√ 5)/2 ≈ 2.618
Using Lemma 3
![Page 40: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/40.jpg)
40
Related Work
Max social cost, parallel links[Mavronicolas, Spirakis,2001], [Czumaj, Vocking, 2002]
Max social cost, Single-Commodity Network[Fotakis, Kontogiannis,Spirakis,2005]
Sum social cost, parallel links[Lucking, Mavronicals, Monien, Rode, 2004]
Splittable, General Network[Roughgarden, Tardos, 2002]
Max, Sum social cost, General Network[Awebuch, Azar, Epstein, 2005][Christodoulou, Koutsoupias, 2005]
![Page 41: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/41.jpg)
41
Results
For linear congestion games
Correlaed PoS = [1.57,1.6]
Correlatted PoA = 2.5
For weighted congestion games
Correlated PoA = (3+√ 5)/2 ≈ 2.618
![Page 42: 1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d445503460f94a21215/html5/thumbnails/42.jpg)
42
The End